This is a famous
paradox, first framed by the Belgian mathematician, Maurice Kraitchik in 1953
and popularised by Doug Hofstadter in 1982. It has been the
subject of papers in philosophical journals in recent years and its current
status is unresolved, though many claim to have solved it. A Google search nets
12,000 hits. I have looked at the first 40 and found none was satisfactory.
Some people were content with showing that switching doesn’t give any benefit,
others even thought that getting a benefit out of switching is an acceptable
result (at least in the open envelope case). Others thought it enough to show
that the probability of 2x is less than that of x/2. Some answers were so
difficult to follow that I had to give them the benefit of the doubt, though in
these cases I could not see how the solution related to the original problem,
ie what part of the reasoning of Method 2 is in error.
To be honest, my solution can and should be criticised for being abstruse, given the incredibly simple nature of the problem. Nor did I reach it unassisted. There are key ideas I picked up from the Net, without which I probably would never have solved the paradox. These are: 1) that one must take the two envelopes from a probability distribution, 2) that any such distribution cannot be uniform, and 3) the example I use in Section 3 (due to Broome), which allows one to see exactly what happens.
Is my
solution the first correct one ever? This is a hubristic claim, but it is
possibly the case. Pride comes before the slip, so I’ll probably have to eat
these words. In the meantime I am a legend in my own lunchtime. Note that the
current solution supersedes the one I posted here earlier. I believe that
solution was inadequate because it only addressed the total gain, not the gain
for each value.
The problem with discussing this paradox is that
anyone who has spent weeks or months working out their solution is unlikely to
feel motivated to put in the hard work to wade through a densely argued
academic paper of 25 pages in order to rigorously evaluate someone else’s work.
Put crudely, writing about it is more fun than reading someone else’s solution.
I have made an honest attempt to understand and evaluate the papers I discuss in
my solution. However, my conclusions are circumscribed by my lack of intellectual
prowess. I also, grudgingly, admit to a bias towards believing that my solution
is the only correct one.
Please let me know what you think of
my solution. My email is "soler@soler 7.com" - this
email address has an extra space - please remove it.
4 July 2009
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